Question

I need to find the derivative. I’m not very good at this so I am answer asap would be sosososo helpful

Answer 1

See below.

Explanation:

First, notice that this is a composition of functions. For instance, let's let
`f(x)=√(x)` and
`g(x)=7x^2+4x+1`. Then, the given equation is essentially
`f(g(x))=√(7x^2+4x+1)`. Thus, we can use the chain rule.

Recall the chain rule:
`(d)/(dx)[f(g(x))]=f'(g(x))\cdot g'(x)`. So, let's find the derivative of each function:

`f(x)=√(x)=x^(1)/(2)`

We can use the Power Rule here:

`f'(x)=(x^(1)/(2))'=(1)/(2)(x^{-(1)/(2) })=(1)/(2√(x))`

Now:

`g(x)=7x^2+4x+1`

Again, use the Power Rule and Sum Rule

`g'(x)=(7x^2+4x+1)'=(7x^2)'+(4x)'+(1)'=14x+4`

Now, we can put them together:

`y'=f'(g(x))\cdot g'(x)=(1)/(2√(g(x))) \cdot (14x+4)`

`y'=(14x+4)/(2√(7x^2+4x+1) ) =(7x+2)/(√(7x^2+4x+1) )`